Approximate and exact controllability of linear difference equations

Yacine Chitour; Guilherme Mazanti; Mario Sigalotti

doi:10.5802/jep.112

Approximate and exact controllability of linear difference equations
[Contrôlabilité approchée et exacte d’équations aux différences linéaires]

Yacine Chitour ¹ ; Guilherme Mazanti ² ; Mario Sigalotti ³

¹ Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506), CNRS — CentraleSupelec — Université Paris-Sud 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France
² Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France
³ Inria & Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions 75005 Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 93-142.

Résumé
Abstract

Cet article traite de la contrôlabilité approchée et exacte de l’équation aux différences linéaire $x (t) = \sum_{j = 1}^{N} A_{j} x (t - Λ_{j}) + B u (t)$ dans $L^{2}$ , avec $x (t) \in ℂ^{d}$ et $u (t) \in ℂ^{m}$ , en s’appuyant sur une formule de représentation de la solution $x$ en termes de la condition initiale, du contrôle $u$ et de coefficients matriciels appropriés. Lorsque $Λ_{1}, \dots, Λ_{N}$ sont commensurables, les contrôlabilités approchée et exacte sont équivalentes et peuvent être caractérisées par un critère de type Kalman. Cet article s’attache à donner des caractérisations des contrôlabilités approchée et exacte sans hypothèse de commensurabilité. Dans le cas d’un système bi-dimensionnel avec deux retards, nous obtenons une caractérisation explicite des contrôlabilités approchée et exacte en termes des paramètres du problème. Pour le cas général, nous prouvons que la contrôlabilité approchée de zéro vers les états constants est équivalente à la contrôlabilité approchée dans $L^{2}$ . Le résultat correspondant à la contrôlabilité exacte est vrai au moins pour les systèmes bi-dimensionnels avec deux retards.

In this paper, we study approximate and exact controllability of the linear difference equation $x (t) = \sum_{j = 1}^{N} A_{j} x (t - Λ_{j}) + B u (t)$ in $L^{2}$ , with $x (t) \in ℂ^{d}$ and $u (t) \in ℂ^{m}$ , using as a basic tool a representation formula for its solution in terms of the initial condition, the control $u$ , and some suitable matrix coefficients. When $Λ_{1}, \dots, Λ_{N}$ are commensurable, approximate and exact controllability are equivalent and can be characterized by a Kalman criterion. This paper focuses on providing characterizations of approximate and exact controllability without the commensurability assumption. In the case of two-dimensional systems with two delays, we obtain an explicit characterization of approximate and exact controllability in terms of the parameters of the problem. In the general setting, we prove that approximate controllability from zero to constant states is equivalent to approximate controllability in $L^{2}$ . The corresponding result for exact controllability is true at least for two-dimensional systems with two delays.

Reçu le : 2019-02-15
Accepté le : 2019-10-04
Publié le : 2019-11-08

MR Zbl

DOI : 10.5802/jep.112

Classification : 39A06, 93B05, 93C65
Keywords: Linear difference equation, delay, approximate controllability, exact controllability
Mot clés : Équation aux différences linéaire, retard, contrôlabilité approchée, contrôlabilité exacte

Affiliations des auteurs :

Yacine Chitour ¹ ; Guilherme Mazanti ² ; Mario Sigalotti ³

¹ Laboratoire des Signaux et Systèmes (L2S, UMR CNRS 8506), CNRS — CentraleSupelec — Université Paris-Sud 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France
² Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay 91405 Orsay, France
³ Inria & Sorbonne Université, Université de Paris, CNRS, Laboratoire Jacques-Louis Lions 75005 Paris, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Yacine Chitour and Guilherme Mazanti and Mario Sigalotti},
     title = {Approximate and exact controllability of linear difference equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     publisher = {\'Ecole polytechnique},
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Yacine Chitour; Guilherme Mazanti; Mario Sigalotti. Approximate and exact controllability of linear difference equations. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 93-142. doi : 10.5802/jep.112. https://jep.centre-mersenne.org/articles/10.5802/jep.112/

Bibliographie
Cité par

[1] C. E. de Avellar & J. K. Hale - “On the zeros of exponential polynomials”, J. Math. Anal. Appl. 73 (1980) no. 2, p. 434-452 | DOI | MR | Zbl

[2] J. W. S. Cassels - An introduction to Diophantine approximation, Cambridge Tracts in Math. and Math. Physics, vol. 45, Cambridge University Press, New York, 1957 | MR | Zbl

[3] Y. Chitour, G. Mazanti & M. Sigalotti - “Stability of non-autonomous difference equations with applications to transport and wave propagation on networks”, Netw. Heterog. Media 11 (2016) no. 4, p. 563-601 | DOI | MR | Zbl

[4] Y. Chitour, G. Mazanti & M. Sigalotti - “Persistently damped transport on a network of circles”, Trans. Amer. Math. Soc. 369 (2017) no. 6, p. 3841-3881 | DOI | MR | Zbl

[5] D. H. Chyung - “On the controllability of linear systems with delay in control”, IEEE Trans. Automatic Control 15 (1970) no. 2, p. 255-257 | DOI | MR

[6] K. L. Cooke & D. W. Krumme - “Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations”, J. Math. Anal. Appl. 24 (1968), p. 372-387 | DOI | MR | Zbl

[7] J.-M. Coron - Control and nonlinearity, Math. Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007 | DOI | MR | Zbl

[8] J.-M. Coron, G. Bastin & B. d’Andréa-Novel - “Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems”, SIAM J. Control Optim. 47 (2008) no. 3, p. 1460-1498 | DOI | MR | Zbl

[9] J.-M. Coron & H.-M. Nguyen - “Dissipative boundary conditions for nonlinear 1-D hyperbolic systems: sharp conditions through an approach via time-delay systems”, SIAM J. Math. Anal. 47 (2015) no. 3, p. 2220-2240 | DOI | MR | Zbl

[10] M. Cruz & J. K. Hale - “Stability of functional differential equations of neutral type”, J. Differential Equations 7 (1970), p. 334-355 | DOI | MR | Zbl

[11] R. Datko - “Linear autonomous neutral differential equations in a Banach space”, J. Differential Equations 25 (1977) no. 2, p. 258-274 | DOI | MR | Zbl

[12] J. Diblík, D. Y. Khusainov & M. Růžičková - “Controllability of linear discrete systems with constant coefficients and pure delay”, SIAM J. Control Optim. 47 (2008) no. 3, p. 1140-1149 | DOI | MR | Zbl

[13] G. F. Franklin, J. D. Powell & M. L. Workman - Digital Control of Dynamic Systems, Addison-Wesley, 1997 | Zbl

[14] E. Fridman, S. Mondié & B. Saldivar - “Bounds on the response of a drilling pipe model”, IMA J. Math. Control Inform. 27 (2010) no. 4, p. 513-526 | DOI | MR | Zbl

[15] I. Gohberg & T. Shalom - “On inversion of square matrices partitioned into nonsquare blocks”, Integral Equations Operator Theory 12 (1989) no. 4, p. 539-566 | DOI | MR | Zbl

[16] J. K. Hale, E. F. Infante & F. S. P. Tsen - “Stability in linear delay equations”, J. Math. Anal. Appl. 105 (1985) no. 2, p. 533-555 | DOI | MR | Zbl

[17] J. K. Hale & S. M. Verduyn Lunel - Introduction to functional-differential equations, Applied Math. Sciences, vol. 99, Springer-Verlag, New York, 1993 | DOI | MR | Zbl

[18] J. K. Hale & S. M. Verduyn Lunel - “Strong stabilization of neutral functional differential equations”, IMA J. Math. Control Inform. 19 (2002) no. 1-2, p. 5-23 | DOI | MR | Zbl

[19] D. Henry - “Linear autonomous neutral functional differential equations”, J. Differential Equations 15 (1974), p. 106-128 | DOI | MR | Zbl

[20] B. Klöss - “The flow approach for waves in networks”, Oper. Matrices 6 (2012) no. 1, p. 107-128 | DOI | MR | Zbl

[21] J.-L. Lions - “Contrôlabilité exacte des systèmes distribués”, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986) no. 13, p. 471-475 | DOI | Zbl

[22] J.-L. Lions - “Exact controllability, stabilization and perturbations for distributed systems”, SIAM Rev. 30 (1988) no. 1, p. 1-68 | DOI | MR

[23] R. Mañé - Ergodic theory and differentiable dynamics, Ergeb. Math. Grenzgeb. (3), vol. 8, Springer-Verlag, Berlin, 1987 | DOI | MR | Zbl

[24] G. Mazanti - “Relative controllability of linear difference equations”, SIAM J. Control Optim. 55 (2017) no. 5, p. 3132-3153 | DOI | MR | Zbl

[25] W. R. Melvin - “Stability properties of functional difference equations”, J. Math. Anal. Appl. 48 (1974), p. 749-763 | DOI | MR | Zbl

[26] W. Michiels, T. Vyhlídal, P. Zítek, H. Nijmeijer & D. Henrion - “Strong stability of neutral equations with an arbitrary delay dependency structure”, SIAM J. Control Optim. 48 (2009) no. 2, p. 763-786 | DOI | MR | Zbl

[27] P. H. A. Ngoc & N. D. Huy - “Exponential stability of linear delay difference equations with continuous time”, Vietnam J. Math. 43 (2015) no. 2, p. 195-205 | DOI | MR | Zbl

[28] D. A. O’Connor & T. J. Tarn - “On stabilization by state feedback for neutral differential-difference equations”, IEEE Trans. Automatic Control 28 (1983) no. 5, p. 615-618 | DOI | MR | Zbl

[29] D. A. O’Connor & T. J. Tarn - “On the function space controllability of linear neutral systems”, SIAM J. Control Optim. 21 (1983) no. 2, p. 306-329 | DOI | MR | Zbl

[30] L. Pandolfi - “Stabilization of neutral functional differential equations”, J. Optimization Theory Appl. 20 (1976) no. 2, p. 191-204 | DOI | MR | Zbl

[31] M. Pospíšil, J. Diblík & M. Fečkan - “On relative controllability of delayed difference equations with multiple control functions”, in Proceedings of the International conference on numerical analysis and applied mathematics 2014 (ICNAAM-2014), vol. 1648, AIP Publishing, 2015, article ID 130001 | DOI

[32] W. Rudin - Real and complex analysis, McGraw-Hill Book Co., New York, 1987 | Zbl

[33] W. Rudin - Functional analysis, International Series in Pure and Applied Math., McGraw-Hill, Inc., New York, 1991 | Zbl

[34] D. Salamon - Control and observation of neutral systems, Research Notes in Math., vol. 91, Pitman, Boston, MA, 1984 | MR | Zbl

[35] M. Slemrod - “Nonexistence of oscillations in a nonlinear distributed network”, J. Math. Anal. Appl. 36 (1971), p. 22-40 | DOI | MR | Zbl

[36] E. D. Sontag - Mathematical control theory. Deterministic finite-dimensional systems, Texts in Applied Math., vol. 6, Springer-Verlag, New York, 1998 | DOI | Zbl

[37] M. Viana - “Ergodic theory of interval exchange maps”, Rev. Mat. Univ. Complut. Madrid 19 (2006) no. 1, p. 7-100 | DOI | MR | Zbl

[38] P. Walters - An introduction to ergodic theory, Graduate Texts in Math., vol. 79, Springer-Verlag, New York-Berlin, 1982 | MR | Zbl

Cité par Sources :